IDEALS AND SUBMODULES OF MULTIPLICATION MODULES

Abstract

Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = 1M. Let M be a non-zero multiplication R-module. Then we prove the following: (1) there exists a bijection: N(M)<TEX>$\bigcap$</TEX>V(ann<TEX>$\_{R}$</TEX>(M))<TEX>$\rightarrow$</TEX>Spec<TEX>$\_{R}$</TEX>(M) and in particular, there exists a bijection: N(M)<TEX>$\bigcap$</TEX>Max(R)<TEX>$\rightarrow$</TEX>Max<TEX>$\_{R}$</TEX>(M), (2) N(M) <TEX>$\bigcap$</TEX> V(ann<TEX>$\_{R}$</TEX>(M)) = Supp(M) <TEX>$\bigcap$</TEX> V(ann<TEX>$\_{R}$</TEX>(M)), and (3) for every ideal I of R, The ideal <TEX>$\theta$</TEX>(M) = <TEX>$\sum$</TEX><TEX>$\_{m</TEX>(Rm :R M) of R has proved useful in studying multiplication modules. We generalize this ideal to prove the following result: Let R be a commutative ring with identity, P <TEX>$\in$</TEX> Spec(R), and M a non-zero R-module satisfying (1) M is a finitely generated multiplication module, (2) PM is a multiplication module, and (3) P<TEX>$^{n}$</TEX>M<TEX>$\neq$</TEX>P<TEX>$^{n+1}$</TEX> for every positive integer n, then <TEX>$\bigcap$</TEX><TEX>$^{</TEX><TEX>$\_{n=1}$</TEX>(P<TEX>$^{n}$</TEX> + ann<TEX>$\_{R}$</TEX>(M)) <TEX>$\in$</TEX> V(ann<TEX>$\_{R}$</TEX>(M)) = Supp(M) <TEX>$\subseteq$</TEX> N(M).